So my plucky halfling street performer bard is whisked away to a magical land and lands in a tropical beach. A six-headed hydra assails the party from the surf. In response, my halfling softens it up with a dirge of doom and successfully enchants it with Charm Monster. I’m level 10, so in theory it is now a nice, friendly hydra for ten days.
So, the question: Can I use Handle Animal or any other mechanic to domesticate the beast and get me a pet-six headed hydra?
The skill Handle Animal on Modifiers on Low Intelligence Non-Animals says, "You can use this skill on a creature with an Intelligence score of 1 or 2 that is not an animal, but the DC of any such check increases by 5. Such creatures have the same limit on tricks known as animals do." Thus you can use the Handle Animal skill with some minor adjustments on a typical 6-headed hydra, a magical beast that has an Intelligence score of 2.
Further, according to the rules, a creature can be domesticated only if the handler rears the creature from birth or infancy. Luckily, the impact of domestication in Pathfinder is small. Domestication means only that folks can make Charisma ability checks (instead of Handle Animal skill checks) to handle (DC 15) or "push" (DC 30) the domesticated nonanimal creature (see here). The game also suggests that domestication typically changes the formerly nondomesticated creature's initial disposition toward everyone from unfriendly to neutral (see here). Any other benefits of domestication are the GM's discretion.1
Hence, whether or not it's the subject of a charm monster spell, if Mr. Noggins, your hydra, is already domesticated,2 and the hydra's disposition toward you is such that it will take orders from you (GM's discretion, but this answer may be of interest), you can guide the hydra's behavior as described immediately above using Cha checks. (I know you're a bard so your Cha is probably pretty high, but those are still some big numbers. Good luck!)
Better, though, is actually having the skill Handle Animal as that means, while Mr. Noggins is affected by your charm monster spell, you can likely make a Handle Animal skill check to have Mr. Noggins perform any trick it knows (DC 10), and you can make a Handle Animal skill check to "push" Mr. Noggins to perform a trick it doesn't know (DC 25). (A hydra found in the wild—like one that emerges unaccompanied from a tropical island's surf—is unlikely to have ever been taught a trick therefore making "pushing" it likely the norm.)
There are at least a few other options while the charm monster spell continues, too.
So, in essence, the GM will have to inform you of the conditions that are required to make the hydra into what you consider a pet. The game makes a distinction between another creature and a creature that's, for example, a cavalier's mount, a druid's animal companion, or a witch's familiar, but the game doesn't make a distinction between another creature and a pet. There are no mechanics for becoming or being another creature's pet.
For that charm monster spell's 10-day duration, you'll have a six-headed hydra who likes you a lot as long as nobody on your side draws a weapon on it.5 One week is enough for a creature skilled in the Handle Animal skill to teach Mr. Noggins a trick—and let me be the first to suggest the entertain trick—, but it would take renewing the charm monster spell so that it lasts 3–6 weeks for a creature skilled in the Handle Animal skill to train Mr. Noggins for a purpose.6
However, if that charm monster spell ends, Mr. Noggins's attitude toward you becomes the attitude it should have toward you had it not been the subject of your charm monster spell for however long it had been the subject of your charm monster spell. With the spell gone, it need not be your trusted friend and ally, and it need not view your words and actions in the most favorable way. What this means will depend on the GM.
In this GM's campaigns, were you cruel to Mr. Noggins while it was the subject of that charm monster spell, Mr. Noggins looks back on your record of cruelty and no longer views it in the most favorable light. In other words, this GM would advise you to have available always a spare charm monster spell or two in case Mr. Noggins is freed, finally sees you as the monster you are, and attacks.
By the same token, were you kind to Mr. Noggins in a meaningful way for a meaningful length of time while it was the subject of that charm monster spell, this GM wouldn't have Mr. Noggins immediately turn on you. On the contrary, if you've been Mr. Noggins primary caregiver for a week or so and you've been generous about it, Mr. Noggins will probably fight on your behalf even if the spell's removed. Still, though, unless Mr. Noggins is again the subject of a charm monster spell, this GM would have Mr. Noggins remain only briefly by your side, leaving you probably when you starting feeding it less than it can hunt down on its own.
1 Domesticating a creature was more important in Dungeons & Dragons, Third Edition but virtually eliminated by the 3.5 revision. Still, legacy text of Third Edition's distinctions remained post-revision and those were ported into Pathfinder sadly without explanation or expansion. (For more see this answer and this answer.)
2 Welcome to Fiora!
3 The charm monster spell redirects to the charm person spell, but the latter makes no allowances for the former being able to affect creatures that possess Intelligence scores of 1 or 2. So this is an option, but it's rarely available given that the subject is a hydra.
4 This, too, comes from the charm person spell. Also, so you know, the skill Perform (act) covers pantomiming but pretty much just for busking not communication.
5 This reader has come to view the ludicrous duration of the charm monster spell as a trap. It never fails. Just when you think you've pleased Mr. Noggins enough after having fed him a cow a day for a week, some pixie or whatever rolls up on day 8 right before breakfast, casts dispel magic, and—bang!—there you are, fighting Mr. Noggins again plus killing some jerk pixie.
6 Being a creature's trainer could come with the advantage of the trainer always being able to use the Handle Animal skill to make a creature the trainer trained perform tricks the trainer taught it. This could be the start of a house rule that mechanically defines the term pet. Also, since you know when your charm monster spell's succeeded (see here), you can cast charm monster again before it expires to extend its duration. Hail hydra.
The handle animal description makes use of the term "animal" several times, which I have seen interpreted as RAW meaning it can only be used on creatures of the animal type. Examples,
You are trained at working with animals, and can teach them tricks, get them to follow your simple commands, or even domesticate them.
And
To push an animal means to get it to perform a task or trick that it doesn’t know but is physically capable of performing. This category also covers making an animal perform a forced march or forcing it to hustle for more than 1 hour between sleep cycles. If the animal is wounded or has taken any nonlethal damage or ability score damage, the DC increases by 2. If your check succeeds, the animal performs the task or trick on its next action.
However, a 6 headed hydra has an intelligence of 2 and so does a riding dog, which can be trained with handle animal. So you can ask your GM if animal in this case includes creatures of animal intelligence. If they agree then you should be able to do it (perhaps with a penalty to the DC's to account for differences between training a normal animal and a magical beast).
Training it while it is charmed seems like a good idea and ought to work as the answers to this question suggest it ought to remember all the training you gave it while it was under the charm effect. Though, you'll have to treat it well to get it to have a friendly/helpful attitude towards you once the charm effect ends.
